New results and algorithms for computing storage functions: The lossless/all-pass cases

Storage functions, which are also the Algebraic Riccati Inequality (ARI) solutions, play an important role in many optimal control/estimation problems. Extreme storage functions are solutions to the corresponding Algebraic Riccati Equation (ARE). While storage functions exist under assumption of dissipativity, a key assumption in formulation of the ARE/ARI is certain `regularity conditions' on the feedthrough term in the input/state/output representation of the system. For example, lossless and all-pass systems do not meet such regularity conditions (nonsingularity of D + DT and I - DTD respectively). And hence the ARE does not exist for such systems. Consequently, computation of storage functions for lossless/all-pass systems is not possible by conventional ARE based methods, and therefore, for such systems, different techniques are needed. In this paper we present three new algorithms for computation of storage functions for lossless/all-pass systems and compare them for numerical accuracy and computational efficiency. Each of the proposed methods presented in this paper comes from different viewpoints. One is linked to the notion of Bezoutian of two polynomials, while another is motivated by Foster realization of LC circuits and the third method is linked to the notion of trajectories of minimal dissipation in the behavioral approach. A comparative study among the three methods shows that the method based on the Bezoutian is the best from both perspectives: computational time and numerical accuracy.

[1]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[2]  Thomas Kailath,et al.  Linear Systems , 1980 .

[3]  Bruno Iannazzo,et al.  Numerical Solution of Algebraic Riccati Equations , 2012, Fundamentals of algorithms.

[4]  H. Pillai,et al.  Algorithm to Compute Minimal Nullspace Basis of a Polynomial Matrix , 2010 .

[5]  Didier Henrion,et al.  An improved Toeplitz algorithm for polynomial matrix null-space computation , 2009, Appl. Math. Comput..

[6]  Harish K. Pillai,et al.  Dissipative Systems Synthesis: a Linear Algebraic Approach , 2005, CDC 2005.

[7]  Jeffrey S. Rosenthal,et al.  Introduction to mathematical systems theory. a behavioral approach [Book Review] , 2002, IEEE Transactions on Automatic Control.

[8]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[9]  P. Benner,et al.  Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey , 2013 .

[10]  Valeria Simoncini,et al.  Computational Methods for Linear Matrix Equations , 2016, SIAM Rev..

[11]  Danny C. Sorensen,et al.  Passivity preserving model reduction via interpolation of spectral zeros , 2003, 2003 European Control Conference (ECC).

[12]  Harry L. Trentelman,et al.  Dissipativity preserving model reduction by retention of trajectories of minimal dissipation , 2009, Math. Control. Signals Syst..

[13]  J. Willems,et al.  On Quadratic Differential Forms , 1998 .

[14]  H. Trentelman,et al.  Dissipative Systems Synthesis: a Linear Algebraic Approach , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[15]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in System and Control Theory , 1994, Studies in Applied Mathematics.

[16]  Peter Benner,et al.  Evaluation of the Linear Matrix Equation Solvers in SLICOT , 2007 .

[17]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .